Question

given rw = wt, uw = ws. prove rstu is a parallelogram.
identify the steps that complete the proof.
statements:

1. rw = wt, uw = ws
2. ∠swt and ∠uwr are vertical angles
3. ∠swt ≅ ∠uwr
4. △swr ≅ △uwt
5. ∠wrs = ∠wtu, ∠wsr = ∠wut
6. ru || st, rt || rs
7. rstu is a parallelogram

reasons:

1. given
2. def of vertical angles
3. *
4. *
5. *
6. converse of alt interior angles theorem
7. def of a parallelogram

Explanation:

Step1: Recall vertical - angle property

Vertical angles are congruent. So, since $\angle SWT$ and $\angle UWR$ are vertical angles, $\angle SWT\cong\angle UWR$ because vertical angles are always congruent.

Step2: Apply Side - Angle - Side (SAS) congruence criterion

We know that $RW = WT$ and $UW=WS$ (given) and $\angle SWT\cong\angle UWR$ (from step 1). So, $\triangle SWR\cong\triangle UWT$ by the Side - Angle - Side (SAS) congruence postulate.

Step3: Use corresponding - parts of congruent triangles

If $\triangle SWR\cong\triangle UWT$, then by the Corresponding Parts of Congruent Triangles are Congruent (CPCTC) theorem, $\angle WRS=\angle WTU$ and $\angle WSR=\angle WUT$.

Answer:

1. Vertical angles are congruent.
2. SAS (Side - Angle - Side)
3. CPCTC (Corresponding Parts of Congruent Triangles are Congruent)